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Pipe stress II

In this article I will discuss allowable displacement stress range equation given in the piping code B31.3 and the way that it is derived. Firstly, let’s see material behaviour under different types of loads. If pipe is loaded with external forces having stress larger than yield stress, pipe after unloading will be deformed due to forming plastic hinge. Please see the figure below.
1-stress-strain-sustein-load
It is evident that pipe stress must be limited to the yield stress, or if we take factor of safety 2/3, than allowable stress is 0.66 of yield stress. Since allowable stress depends of temperature, we differentiate allowable stress in the cold state Sc and allowable stress at the hot state Sh. If pipe is loaded with thermal load having stress larger than yield stress, pipe after unloading will return in its original position, but it will have residual stress (sign is opposite form thermal load). This residual stress is the hot allowable stress or Sh. Please see the figure below.
2-stress-strain-thermal-load
Please note that pipe in the second cycle can sustain stress equal to Sc+Sh, which it is in piping terminology called thermal stress range. A pipe under thermal load will experience stress larger than yield; in fact this stress is 2 times of yield stress, which has been illustrated on the figure below.
3-stress-range-piping
Since piping system is exposed to external and thermal load, combine stress from external load and thermal load cannot exceed sum of the cold and hot yield stress, having limited stress from sustain load to hot yield stress. Please see the figure below.
4-allowable-displacement-stress-range
Eq.1 is derived taking cold (hot) yield stress equal to 3/2 of allowable cold (hot) stress, and applying factor of safety (FOS). It should be noted that factor of safety has been applied two times; reducing 1.5 to 1.25; and taking allowable stress as 2/3 of yield stress. Eq.2 can be derived, rearranging Eq.1. Please see the figure below.
5-allowable-displacement-stress-range
Since Sh is reserved for sustain load, it must be taken out of Eq.2. Please see the figure below.
6-allowable-displacement-stress-range
Since piping system goes thru cycles, fatigue stress also must be taken into consideration. Loading cycles might comes from temperature oscillation due to process requirement or during plant maintenance shut down when piping system will be unloaded. Fatigue stress is taken into consideration by multiplying allowable stress with fatigue factor f which is called stress range factor in the piping terminology. By applying stress range factor f, final equation become
7-allowable-displacement-stress-range
Note that thermal stress is caused by thermal strain. When stress due to thermal load reach yield stress, material will flow by itself and stress caused by thermal loading will diminish. This stress is called self limiting stress.In this state pipe can take thermal load having the stress that is 2 times of yield stress.